An army of orcs has marched upon King Amit's stronghold, setting up camp a safe distance away (or so they think!). What the orcs don't know is that the king and his engineers have spent the past decade developing advanced catapulting technology. King Amit has created a map to help him compute the distance from each of his two best catapults to the enemy orc camp (shown below). To what distance should King Amit calibrate catapult $B$ to fire? Do not round during your calculations. Round your final answer to the nearest meter. $96^\circ$ $\,58^\circ$ $\;145\,\text{m}$ $ ?$ $A$ $B$ $\text{Enemy orc camp}$
Solution: Converting the problem into geometrical terms Our problem can be modeled by the following triangle $\triangle ABC$, where we want to find $BC=d$. Because the interior angles of a triangle add to $180^\circ$, we know that $\angle C=26^\circ$. $96^\circ$ $\,58^\circ$ $26^\circ$ $\;145\,\text{m}$ $d$ $A$ $B$ $C$ Since we are given one side length and all angle measures, we can use the law of sines. Using the law of sines $\begin{aligned} \dfrac{\sin(C)}{AB}&=\dfrac{\sin(A)}{BC}\\\\ \dfrac{\sin(26^\circ)}{145} &= \dfrac{\sin(96^\circ)}{d} \gray{\text{Substitute}} \\\\ d \cdot \sin(26^\circ) &= 145 \cdot \sin(96^\circ) \\\\ d &= \dfrac{145 \cdot \sin(96^\circ) }{\sin(26^\circ) } \\\\ d &\approx 329 \,\text{m} \end{aligned}$ The answer Amit should calibrate catapult $B$ to fire $329 \,\text{m}$.